How to add a phone to AppleCare?

Start with Katana-San's advice. I'll add that some people want to know if they can add a spouse's device to their AppleCare One plan. Typically the answer is "no". From:

AppleCare - Apple

Can I add my family’s devices to my AppleCare One plan?

AppleCare One plans can cover devices that are on the same Apple Account as the subscriber.

So unless your wife is also using your Apple Account (which is a bad idea), she is on a different account.

Hello~ We have no way of knowing if your wife’s iPhone is eligible. However, you may click on below to check…

Add AppleCare coverage to your Apple device - Apple Support

~Katana-San~

You replied to Katana-San. Did you mean that answer was not the one you were not looking for? You only get a certain time window of eligibility for AppleCare.